Don't use a negative sidebearing! Treat lowercase dcaron/lcaron just like the f: nnďánnfánn.

For uppercase isn't the caron on D the upside down circumflex one? Shouldn't be any problems with crashing.

Treat lowercase dcaron/lcaron just like the f
Don't you usually use a negative sidebearing for f?

Comparing this question to an f is erroneous for a variety of reasons. The f has a wide range of form depending on the languages and technologies you expect to design to. Newspapers with no kerning need one solution- a narrow f. Users with access to kerning, ligatures and contextual shapes can use another solutions. If the font is just for a language with no clashes between f and diacritic mark then even more extravagant shapes are possible. In the case of the dcaron and lcaron the range of desirable variation is much much smaller. The f is very big and isn't going to be misread as easily as a small mark could be if an overlap occurs.

I suggest that it is nearly always best to err on the side of using just space to make your worst case spacing scenario . I say this because it is practical - you can kern your way out out. But also because my reading of the science on the topic suggest that clashes are worse for reading that gaps are. People tolerate gaps really well. The shape of the Caron is relevant too. It is tall and thin. It far more susceptible to being misread/unidentified/ignored when placed next to an overlapping feature than other kinds of marks would be.

You might also look at a cross section of fonts you respect.

You might also ask David Březina at Rosetta Type http://rosettatype.com/ .

What about some example images so we can see the context as well?

Assuming no kerning:

Statistically, the vast majority of letter combinations involving dcaron, lcaron and tcaron are better served by a negative right sidebearing, while allowing occasional failure.

Also bear in mind words that end with dcaron, lcaron and tcaron, being followed by a space or (baseline) punctuation mark. Again, negative right sidebearing works better.

Sorry for causing a mess. Sometimes I can't communicate properly and regret later.

I meant that carons are the same type of "design bug" as the f. So for "environments with no kerning" the best way is just to accept the gap, as we accept the gap after f. If not (I think of spacing next to the x-height letters), half of accents and all ascenders will crash with the caron. That's my "negative sidebearing".
I guess, nnfn/lcaron/adieresis/n/n in Adobe Caslon Pro - the overall rhythm, and specially the rhythm of accents, present themselves quiet well .